AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Average Error: 27.1 → 12.6

Time: 7.5s

Precision: 64

Math

\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.9701480721904167 \cdot 10^{207} \lor \neg \left(y \le 2.6574689117362167 \cdot 10^{63}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{\frac{\left(x + t\right) + y}{t + y}} + \frac{\left(x + y\right) \cdot z - y \cdot b}{\left(x + t\right) + y}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (((double) (((double) (((double) (x + y)) * z)) + ((double) (((double) (t + y)) * a)))) - ((double) (y * b)))) / ((double) (((double) (x + t)) + y))));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((y <= -1.9701480721904167e+207) || !(y <= 2.6574689117362167e+63))) {
		VAR = ((double) (((double) (a + z)) - b));
	} else {
		VAR = ((double) (((double) (((double) (a * 1.0)) / ((double) (((double) (((double) (x + t)) + y)) / ((double) (t + y)))))) + ((double) (((double) (((double) (((double) (x + y)) * z)) - ((double) (y * b)))) / ((double) (((double) (x + t)) + y))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	27.1
Target	11.0
Herbie	12.6

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -1.9701480721904167e+207 or 2.6574689117362167e+63 < y

   1. Initial program 45.0

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
   2. Using strategy rm

   3. Applied clear-num 45.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}}\]

   4. Taylor expanded around 0 13.0

      \[\leadsto \color{blue}{\left(a + z\right) - b}\]

   if -1.9701480721904167e+207 < y < 2.6574689117362167e+63

   1. Initial program 20.1

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
   2. Using strategy rm

   3. Applied clear-num 20.2

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}}\]
   4. Using strategy rm

   5. Applied div-inv 20.3

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x + t\right) + y\right) \cdot \frac{1}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}}\]

   6. Applied add-cube-cbrt 20.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\left(x + t\right) + y\right) \cdot \frac{1}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\]

   7. Applied times-frac 20.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left(x + t\right) + y} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}}\]

   8. Simplified 20.2

      \[\leadsto \color{blue}{\frac{1}{\left(x + t\right) + y}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\]

   9. Simplified 20.2

      \[\leadsto \frac{1}{\left(x + t\right) + y} \cdot \color{blue}{\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right)}\]
   10. Using strategy rm

   11. Applied +-commutative 20.2

       \[\leadsto \frac{1}{\left(x + t\right) + y} \cdot \left(\color{blue}{\left(\left(t + y\right) \cdot a + \left(x + y\right) \cdot z\right)} - y \cdot b\right)\]

   12. Applied associate--l+ 20.2

       \[\leadsto \frac{1}{\left(x + t\right) + y} \cdot \color{blue}{\left(\left(t + y\right) \cdot a + \left(\left(x + y\right) \cdot z - y \cdot b\right)\right)}\]

   13. Applied distribute-lft-in 20.2

       \[\leadsto \color{blue}{\frac{1}{\left(x + t\right) + y} \cdot \left(\left(t + y\right) \cdot a\right) + \frac{1}{\left(x + t\right) + y} \cdot \left(\left(x + y\right) \cdot z - y \cdot b\right)}\]

   14. Simplified 20.1

       \[\leadsto \color{blue}{\frac{a \cdot \left(t + y\right)}{\left(x + t\right) + y}} + \frac{1}{\left(x + t\right) + y} \cdot \left(\left(x + y\right) \cdot z - y \cdot b\right)\]

   15. Simplified 20.1

       \[\leadsto \frac{a \cdot \left(t + y\right)}{\left(x + t\right) + y} + \color{blue}{\frac{\left(x + y\right) \cdot z - y \cdot b}{\left(x + t\right) + y}}\]
   16. Using strategy rm

   17. Applied *-un-lft-identity 20.1

       \[\leadsto \frac{a \cdot \color{blue}{\left(1 \cdot \left(t + y\right)\right)}}{\left(x + t\right) + y} + \frac{\left(x + y\right) \cdot z - y \cdot b}{\left(x + t\right) + y}\]

   18. Applied associate-*r* 20.1

       \[\leadsto \frac{\color{blue}{\left(a \cdot 1\right) \cdot \left(t + y\right)}}{\left(x + t\right) + y} + \frac{\left(x + y\right) \cdot z - y \cdot b}{\left(x + t\right) + y}\]

   19. Applied associate-/l* 12.4

       \[\leadsto \color{blue}{\frac{a \cdot 1}{\frac{\left(x + t\right) + y}{t + y}}} + \frac{\left(x + y\right) \cdot z - y \cdot b}{\left(x + t\right) + y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.9701480721904167 \cdot 10^{207} \lor \neg \left(y \le 2.6574689117362167 \cdot 10^{63}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{\frac{\left(x + t\right) + y}{t + y}} + \frac{\left(x + y\right) \cdot z - y \cdot b}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))